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Dynamical properties of certain continuous self maps of the Cantor set

机译：Cantor集的某些连续自映射的动力学性质

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Given a dynamical system (X, f) with X a compact metric space and a free ultrafilter p on N, we define f~p(x) = p-lim_(n→∞) f~n(x) for all x ∈ X. It was proved by A. Blass (1993) that x ∈ X is recurrent iff there is p ∈ N~* = β(N) N such that f~p(x) = x. This suggests to consider those points x ∈ X for which f~p(x) = x for some p ∈ N~*, which are called p-recurrent. We shall give an example of a recurrent point which is not p-recurrent for several p ∈ N~*. Also, A. Blass proved that two points x, y ∈ X are proximal iff there is p ∈ N~* such that f~p(x) = f~p(y) (in this case, we say that x and y are p-proximal). We study the properties of the p-proximal points of the following continuous self maps of the Cantor set: For an arbitrary function f : N → N, we define σ_f : {0,1}~N → {0,1}~N by σ_f(x)(k)=x(f(k)) for every k ∈ N and for every x ∈ {0,1}~N (the shift map on {0,1}~N is obtained by the function k_1 → k + 1). Let E(X) denote the Ellis semigroup of the dynamical system (X, f). We prove that if f : N → N is a function with at least one infinite orbit, then E({0,1}~N, σ_f) is homeomor-phic to β(N). Two functions g, h : N → N are defined so that E({0,1}~N, σ_g) is homeomor-phic to the Cantor set, and E({0,1}~N,σ_h) is the one-point compactification of N with the discrete topology.
机译：给定一个动力学系统（X，f），其中X具有一个紧凑的度量空间并且在N上具有一个自由超滤器p，我们针对所有x∈定义f〜p（x）= p-lim_（n→∞）f〜n（x） X. A. Blass（1993）证明x∈X是递归的，前提是p∈N〜* =β（N）N使得f〜p（x）= x。这建议考虑对于某些p∈N〜*而言f〜p（x）= x的那些点x∈X，称为p递归。我们将给出一个递归点的示例，该递归点对于几个p∈N〜*都不是p递归的。同样，A.Blass证明了x，y∈X的两个点是近点，前提是存在p∈N〜*使得f〜p（x）= f〜p（y）（在这种情况下，我们说x和y是p近端）。我们研究以下Cantor集的连续自映射的p近点的性质：对于任意函数f：N→N，我们定义σ_f：{0,1}〜N→{0,1}〜N通过σ_f（x）（k）= x（f（k））对于每个k∈N和每个x∈{0,1}〜N（{0,1}〜N的位移图通过函数获得k_1→k + 1）。令E（X）表示动力学系统（X，f）的Ellis半群。我们证明，如果f：N→N是具有至少一个无限轨道的函数，则E（{0,1}〜N，σ_f）对β（N）是同胚的。定义了两个函数g，h：N→N，以使E（{0,1}〜N，σ_g）对Cantor集是同位的，而E（{0,1}〜N，σ_h）是一个离散拓扑的N点压缩。

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